Schumacher has required constant care since he sustained severe head injuries in an accident while skiing in 2013.
Hamilton took a big step closer to matching Schumacher's record of seven titles too after his teammate Valtteri Bottas failed to finish with an engine problem. Hamilton now leads the championship by 69 points with six races to go.
Hamilton took the win by 4.5 seconds from Red Bull's Max Verstappen, with third for Daniel Ricciardo in Renault's first podium finish since 2011.
Hamilton started faster than Bottas and had the inside line into the hairpin bend at turn one. He forced Bottas wide over a runoff area, but the Finn held on to take the lead back from Hamilton on the inside of the second corner.
Hamilton took the lead back on lap 13 when Bottas braked late and ran wide. The Finn's race was soon over when he complained of having "no power" and stopped in the pits with what the team said was a problem with his hybrid power unit.
Lando Norris was running third and chasing a podium finish for McLaren before his car started losing power. He eventually pulled over with a fire inside the car's right side pod, bringing out the safety car. Verstappen was slow to react at the restart and rather than threatening Hamilton, he nearly lost second to Ricciardo.
Sergio Perez took fourth for Racing Point, ahead of McLaren's Carlos Sainz in fifth and Pierre Gasly for AlphaTauri.
It was another long race for Ferrari in a troubled season for the Italian team. Charles Leclerc started fourth but lacked the pace to hold the position and finished seventh while Sebastian Vettel trailed in 11th after a spin.
A day after getting the call to drive for Racing Point as a late replacement for the unwell Lance Stroll, Nico Hulkenberg stormed through the field from last on the grid to take eighth. Romain Grosjean was ninth for Haas and Antonio Giovinazzi took the final point in 10th for Alfa Romeo.
Another record fell as Kimi Raikkonen started his 323rd race, beating the mark set by Rubens Barrichello from 1993 through 2011. Raikkonen placed 12th after he picked up a time penalty for colliding with George Russell and sending the Williams driver into a spin. Russell retired with a puncture soon after.
Raikkonen's car threw up gravel shortly after the start, sending stones flying at Grosjean's car behind him. Grosjean said he'd been struck in the hand and wondered if he had a broken finger.
It was the first F1 race since 2013 at the Nürburgring in Germany and was named the Eifel Grand Prix after a nearby mountain range.
After Hamilton matched Schumacher's record in front of a German crowd at a track where Schumacher won five times, he can surpass it on a track where F1 has never raced before. The Algarve circuit in Portimao, Portugal, hosts its first race on October 25 as an extra round on the calendar amid the coronavirus pandemic.